Finite Group Databasebeta

← C3×#25661 #10644268

Subgroups — C3×#25661 #10644268

Order 768.

1048 conjugacy classes of subgroups.

The subgroup lattice has 530 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#556C211yesmaximal, characteristic
2C2×C3×#489C211yesmaximal, characteristic
3C2×C3×#477C211yesmaximal, characteristic, complemented
4C3×#2546C211yesmaximal, characteristic, complemented
5C3×#2043C212yesmaximal, complemented
6C3×#2042C212yesmaximal
7C3×#1985C211yesmaximal, characteristic, complemented
8C3×#1983C211yesmaximal, characteristic
9C3×#1863C212yesmaximal, complemented
10C3×#1862C212yesmaximal
11C3×#1763C211yesmaximal, characteristic, complemented
12#25661C311yesmaximal, characteristic, direct product factor, Sylow
13C3×#556—24no—
14C3×#556—24no—
15C2×C3×C4⋊Q8—21no—
16C2×C3×C4⋊Q8C2211yescharacteristic
17C2×C3×C4⋊Q8C2211yescharacteristic
18C2×C3×#126—21no—
19C2×C3×#126—21no—
20C2×C3×#124C2211yescharacteristic
21C2×C3×#124C2211yescharacteristic
22C2×C3×C22⋊Q8—21no—
23C2×C3×C4×Q8C2211yescharacteristic
24C2×C3×C4×Q8C2211yescharacteristic
25C2×C3×C4×Q8C2211yescharacteristic
26C2×C3×C4×D4C2211yescharacteristic
27C2×C3×#117—21no—
28C3×#489—24no—
29C3×#477—24no—
30C2×C3×#107C2212yes—
31C2×C3×C4⋊C8C2211yescharacteristic
32C2×C3×#103C2212yes—
33C2×C3×#103C2212yes—
34C2×C3×#103C2212yes—
35C2×C3×#103C2212yes—
36C2×C3×#102C2212yes—
37C2×C3×#102C2212yes—
38C2×C3×#97C2211yescharacteristic
39C3×#398—21no—
40C3×#398—21no—
41C3×#391—21no—
42C3×#389C2212yes—
43C3×#389C2212yes—
44C3×C4×C4⋊C4C2211yescharacteristic
45C3×#340C2212yes—
46C3×#336C2212yes—
47C3×#336C2212yes—
48C2×#556C2×C311yescharacteristic
49C2×#489C2×C311yescharacteristic
50C2×#477C2×C311yescharacteristic, complemented
51#2546C2×C311yescharacteristic, complemented
52#2043C2×C312yescomplemented
53#2042C2×C312yes—
54#1985C2×C311yescharacteristic, complemented
55#1983C2×C311yescharacteristic
56#1863C2×C312yescomplemented
57#1862C2×C312yes—
58#1763C2×C311yescharacteristic, complemented
59C22×C3×Q8—21no—
60C22×C3×Q8D411yescharacteristic
61C22×C3×Q8C2311yescharacteristic
62C22×C3×Q8C2311yescharacteristic
63C22×C3×D4C2311yescharacteristic
64C23×C3×C4—21no—
65C22×C3×C8C2312yes—
66C22×C3×C8C2312yes—
67C3×C4⋊Q8—22no—
68C3×C4⋊Q8—22no—
69C3×C4⋊Q8—22no—
70C3×C4⋊Q8—22no—
71C3×C4⋊Q8—44no—
72C3×#126—28no—
73C3×#126—28no—
74C3×#124—22no—
75C3×#124—22no—
76C3×#124—22no—
77C3×#124D414yes—
78C3×C22⋊Q8—28no—
79C3×C4×Q8—22no—
80C3×C4×Q8—22no—
81C3×C4×Q8—22no—
82C3×C4×Q8—22no—
83C3×C4×Q8—22no—
84C3×C4×Q8D414yes—
85C3×C4×D4—22no—
86C3×C4×D4—22no—
87C3×#117—28no—
88C2×C3×C4⋊C4—21no—
89C2×C3×C4⋊C4—21no—
90C2×C3×C4⋊C4—21no—
91C2×C3×C4⋊C4—21no—
92C2×C3×C4⋊C4—22no—
93C2×C3×C4⋊C4—22no—
94C2×C3×C4⋊C4—21no—
95C2×C3×C4⋊C4D411yescharacteristic
96C2×C3×C4⋊C4D411yescharacteristic
97C2×C3×C4⋊C4D411yescharacteristic
98C2×C3×C4⋊C4C2312yes—
99C2×C3×C4⋊C4C2311yescharacteristic
100C2×C3×C4⋊C4C2312yes—
101C2×C3×C22⋊C4—21no—
102C2×C3×C22⋊C4—21no—
103C2×C3×C22⋊C4—21no—
104C2×C3×C22⋊C4—21no—
105C2×C3×C22⋊C4—21no—
106C2×C3×C42—21no—
107C2×C3×C42—21no—
108C2×C3×C42—21no—
109C2×C3×C42C2311yescharacteristic
110C2×C3×C42C2311yescharacteristic
111C2×C3×C42C2311yescharacteristic
112C3×#107—24no—
113C3×C4⋊C8—22no—
114C3×#103—24no—
115C3×#103—24no—
116C3×#103—24no—
117C3×#103—24no—
118C3×#102—24no—
119C3×#102—24no—
120C3×#97—22no—
121C3×#95—21no—
122C3×#95—22no—
123C3×#95—22no—
124#556—24no—
125#556—24no—
126C2×C4⋊Q8—21no—
127C2×C4⋊Q8C22×C311yescharacteristic
128C2×C4⋊Q8C22×C311yescharacteristic
129C2×#126—21no—
130C2×#126—21no—
131C2×#124C22×C311yescharacteristic
132C2×#124C22×C311yescharacteristic
133C2×C22⋊Q8—21no—
134C2×C4×Q8C22×C311yescharacteristic
135C2×C4×Q8C22×C311yescharacteristic
136C2×C4×Q8C22×C311yescharacteristic
137C2×C4×D4C22×C311yescharacteristic
138C2×#117—21no—
139#489—24no—
140#477—24no—
141C2×#107C22×C312yes—
142C2×C4⋊C8C22×C311yescharacteristic
143C2×#103C22×C312yes—
144C2×#103C22×C312yes—
145C2×#103C22×C312yes—
146C2×#103C22×C312yes—
147C2×#102C22×C312yes—
148C2×#102C22×C312yes—
149C2×#97C22×C311yescharacteristic
150#398—21no—
151#398—21no—
152#391—21no—
153#389C22×C312yes—
154#389C22×C312yes—
155C4×C4⋊C4C22×C311yescharacteristic
156#340C22×C312yes—
157#336C22×C312yes—
158#336C22×C312yes—
159C24×C3—21no—
160C2×C3×Q8—22no—
161C2×C3×Q8—22no—
162C2×C3×Q8—21no—
163C2×C3×Q8—21no—
164C2×C3×Q8—22no—
165C2×C3×Q8—22no—
166C2×C3×Q8—22no—
167C2×C3×Q8—22no—
168C2×C3×Q8—41no—
169C2×C3×Q8—42no—
170C2×C3×Q8—42no—
171C2×C3×Q8—42no—
172C2×C3×Q8C2×D412yes—
173C2×C3×Q8C2×D412yes—
174C2×C3×D4—21no—
175C2×C3×D4—22no—
176C2×C3×D4—22no—
177C2×C3×D4C2×D412yes—
178C22×C3×C4—24no—
179C22×C3×C4—22no—
180C22×C3×C4—21no—
181C22×C3×C4—21no—
182C22×C3×C4—21no—
183C22×C3×C4—22no—
184C22×C3×C4—42no—
185C22×C3×C4—42no—
186C22×C3×C4C2×D411yescharacteristic
187C22×C3×C4C2×D411yescharacteristic
188C22×C3×C4C2×D411yescharacteristic
189C22×C3×C4C2×D411yescharacteristic
190C22×C3×C4C2×D411yescharacteristic
191C22×C3×C4C2×D411yescharacteristic
192C22×C3×C4C2411yescharacteristic
193C2×C3×C8—22no—
194C2×C3×C8—22no—
195C2×C3×C8—22no—
196C2×C3×C8—22no—
197C2×C3×C8—22no—
198C2×C3×C8—22no—
199C3×C4⋊C4—24no—
200C3×C4⋊C4—24no—
201C3×C4⋊C4—24no—
202C3×C4⋊C4—24no—
203C3×C4⋊C4—22no—
204C3×C4⋊C4—22no—
205C3×C4⋊C4—22no—
206C3×C4⋊C4—21no—
207C3×C4⋊C4—22no—
208C3×C4⋊C4—22no—
209C3×C4⋊C4—22no—
210C3×C4⋊C4—44no—
211C3×C4⋊C4—42no—
212C3×C4⋊C4—44no—
213C3×C4⋊C4C2×D412yes—
214C3×C4⋊C4#4114yes—
215C3×C22⋊C4—24no—
216C3×C22⋊C4—24no—
217C3×C22⋊C4—24no—
218C3×C22⋊C4—24no—
219C3×C22⋊C4—24no—
220C3×C42—24no—
221C3×C42—24no—
222C3×C42—21no—
223C3×C42—21no—
224C3×C42—21no—
225C3×C42—21no—
226C3×C42—42no—
227C3×C42C2×D412yes—
228C3×C42C2×D412yes—
229C22×Q8—21no—
230C22×Q8C3×D411yescharacteristic
231C22×Q8C23×C311yescharacteristic
232C22×Q8C23×C311yescharacteristic
233C22×D4C23×C311yescharacteristic
234C23×C4—21no—
235C22×C8C23×C312yes—
236C22×C8C23×C312yes—
237C4⋊Q8—22no—
238C4⋊Q8—22no—
239C4⋊Q8—22no—
240C4⋊Q8—22no—
241C4⋊Q8—44no—
242#126—28no—
243#126—28no—
244#124—22no—
245#124—22no—
246#124—22no—
247#124C3×D414yes—
248C22⋊Q8—28no—
249C4×Q8—22no—
250C4×Q8—22no—
251C4×Q8—22no—
252C4×Q8—22no—
253C4×Q8—22no—
254C4×Q8C3×D414yes—
255C4×D4—22no—
256C4×D4—22no—
257#117—28no—
258C2×C4⋊C4—21no—
259C2×C4⋊C4—21no—
260C2×C4⋊C4—21no—
261C2×C4⋊C4—21no—
262C2×C4⋊C4—22no—
263C2×C4⋊C4—22no—
264C2×C4⋊C4—21no—
265C2×C4⋊C4C3×D411yescharacteristic
266C2×C4⋊C4C3×D411yescharacteristic
267C2×C4⋊C4C3×D411yescharacteristic
268C2×C4⋊C4C23×C312yes—
269C2×C4⋊C4C23×C311yescharacteristic
270C2×C4⋊C4C23×C312yes—
271C2×C22⋊C4—21no—
272C2×C22⋊C4—21no—
273C2×C22⋊C4—21no—
274C2×C22⋊C4—21no—
275C2×C22⋊C4—21no—
276C2×C42—21no—
277C2×C42—21no—
278C2×C42—21no—
279C2×C42C23×C311yescharacteristic
280C2×C42C23×C311yescharacteristic
281C2×C42C23×C311yescharacteristic
282#107—24no—
283C4⋊C8—22no—
284#103—24no—
285#103—24no—
286#103—24no—
287#103—24no—
288#102—24no—
289#102—24no—
290#97—22no—
291#95—21no—
292#95—22no—
293#95—22no—
294C23×C3—22no—
295C23×C3—42no—
296C23×C3—41no—
297C23×C3—41no—
298C23×C3—42no—
299C23×C3C22×D411yescharacteristic
300C3×Q8—22no—
301C3×Q8—22no—
302C3×Q8—24no—
303C3×Q8—44no—
304C3×Q8—41no—
305C3×Q8—41no—
306C3×Q8—42no—
307C3×Q8—84no—
308C3×D4—24no—
309C3×D4—41no—
310C3×D4—41no—
311C2×C3×C4—21no—
312C2×C3×C4—21no—
313C2×C3×C4—21no—
314C2×C3×C4—21no—
315C2×C3×C4—22no—
316C2×C3×C4—21no—
317C2×C3×C4—22no—
318C2×C3×C4—21no—
319C2×C3×C4—22no—
320C2×C3×C4—22no—
321C2×C3×C4—21no—
322C2×C3×C4—24no—
323C2×C3×C4—21no—
324C2×C3×C4—21no—
325C2×C3×C4—22no—
326C2×C3×C4—22no—
327C2×C3×C4—48no—
328C2×C3×C4—42no—
329C2×C3×C4—42no—
330C2×C3×C4—42no—
331C2×C3×C4—41no—
332C2×C3×C4—41no—
333C2×C3×C4—42no—
334C2×C3×C4—42no—
335C2×C3×C4—42no—
336C2×C3×C4—42no—
337C2×C3×C4#12012yes—
338C2×C3×C4#12012yes—
339C2×C3×C4AGL(1,ℤ/8)12yes—
340C2×C3×C4C22×D411yescharacteristic
341C2×C3×C4C22×D411yescharacteristic
342C2×C3×C4C2×#4112yes—
343C2×C3×C42−1+412yes—
344C3×C8—42no—
345C3×C8—42no—
346C24—21no—
347C2×Q8—22no—
348C2×Q8—22no—
349C2×Q8—21no—
350C2×Q8—21no—
351C2×Q8—22no—
352C2×Q8—22no—
353C2×Q8—22no—
354C2×Q8—22no—
355C2×Q8—41no—
356C2×Q8—42no—
357C2×Q8—42no—
358C2×Q8—42no—
359C2×Q8C2×C3×D412yes—
360C2×Q8C2×C3×D412yes—
361C2×D4—21no—
362C2×D4—22no—
363C2×D4—22no—
364C2×D4C2×C3×D412yes—
365C22×C4—24no—
366C22×C4—22no—
367C22×C4—21no—
368C22×C4—21no—
369C22×C4—21no—
370C22×C4—22no—
371C22×C4—42no—
372C22×C4—42no—
373C22×C4C2×C3×D411yescharacteristic
374C22×C4C2×C3×D411yescharacteristic
375C22×C4C2×C3×D411yescharacteristic
376C22×C4C2×C3×D411yescharacteristic
377C22×C4C2×C3×D411yescharacteristic
378C22×C4C2×C3×D411yescharacteristic
379C22×C4C24×C311yescharacteristic
380C2×C8—22no—
381C2×C8—22no—
382C2×C8—22no—
383C2×C8—22no—
384C2×C8—22no—
385C2×C8—22no—
386C4⋊C4—24no—
387C4⋊C4—24no—
388C4⋊C4—24no—
389C4⋊C4—24no—
390C4⋊C4—22no—
391C4⋊C4—22no—
392C4⋊C4—22no—
393C4⋊C4—21no—
394C4⋊C4—22no—
395C4⋊C4—22no—
396C4⋊C4—22no—
397C4⋊C4—44no—
398C4⋊C4—42no—
399C4⋊C4—44no—
400C4⋊C4C2×C3×D412yes—
401C4⋊C4C3×#4114yes—
402C22⋊C4—24no—
403C22⋊C4—24no—
404C22⋊C4—24no—
405C22⋊C4—24no—
406C22⋊C4—24no—
407C42—24no—
408C42—24no—
409C42—21no—
410C42—21no—
411C42—21no—
412C42—21no—
413C42—42no—
414C42C2×C3×D412yes—
415C42C2×C3×D412yes—
416C22×C3—42no—
417C22×C3—42no—
418C22×C3—42no—
419C22×C3—82no—
420C22×C3—82no—
421C22×C3C2×#12011yescharacteristic
422C22×C3#54812yes—
423C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
424C22×C3#57511yescharacteristic
425C22×C3#57812yes—
426C3×C4—22no—
427C3×C4—22no—
428C3×C4—22no—
429C3×C4—44no—
430C3×C4—42no—
431C3×C4—42no—
432C3×C4—41no—
433C3×C4—42no—
434C3×C4—41no—
435C3×C4—41no—
436C3×C4—82no—
437C3×C4#53211yescharacteristic
438C3×C4#53611yescharacteristic
439C3×C4#54712yes—
440C23—22no—
441C23—41no—
442C23—41no—
443C23—42no—
444C23—42no—
445C23C22×C3×D411yescharacteristic
446Q8—22no—
447Q8—22no—
448Q8—24no—
449Q8—44no—
450Q8—41no—
451Q8—41no—
452Q8—42no—
453Q8—84no—
454D4—24no—
455D4—41no—
456D4—41no—
457C2×C4—21no—
458C2×C4—21no—
459C2×C4—21no—
460C2×C4—22no—
461C2×C4—21no—
462C2×C4—21no—
463C2×C4—21no—
464C2×C4—22no—
465C2×C4—22no—
466C2×C4—21no—
467C2×C4—22no—
468C2×C4—21no—
469C2×C4—22no—
470C2×C4—22no—
471C2×C4—21no—
472C2×C4—24no—
473C2×C4—48no—
474C2×C4—42no—
475C2×C4—41no—
476C2×C4—41no—
477C2×C4—42no—
478C2×C4—42no—
479C2×C4—42no—
480C2×C4—42no—
481C2×C4—42no—
482C2×C4—42no—
483C2×C4C3×#12012yes—
484C2×C4C3×#12012yes—
485C2×C4C3×AGL(1,ℤ/8)12yes—
486C2×C4C22×C3×D411yescharacteristic
487C2×C4C22×C3×D411yescharacteristic
488C2×C4C2×C3×#4112yes—
489C2×C4C3×2−1+412yes—
490C8—42no—
491C8—42no—
492C2×C3—82no—
493C2×C3#243211yescharacteristic
494C2×C3#300611yescharacteristic
495C2×C3#301111yescharacteristic
496C2×C3#336712yes—
497C2×C3#337712yes—
498C22—42no—
499C22—42no—
500C22—42no—
501C22—82no—
502C22—82no—
503C22C2×C3×#12011yescharacteristic
504C22C3×#54812yes—
505C22C2×C3×AGL(1,ℤ/8)11yescharacteristic
506C22C3×#57511yescharacteristic
507C22C3×#57812yes—
508C4—22no—
509C4—22no—
510C4—22no—
511C4—42no—
512C4—42no—
513C4—41no—
514C4—42no—
515C4—41no—
516C4—41no—
517C4—44no—
518C4—82no—
519C4C3×#53211yescharacteristic
520C4C3×#53611yescharacteristic
521C4C3×#54712yes—
522C3#2566111yescharacteristic, direct product factor, Sylow
523C2—82no—
524C2C3×#243211yescharacteristic
525C2C3×#300611yescharacteristic
526C2C3×#301111yescharacteristic
527C2C3×#336712yes—
528C2C3×#337712yes—
subgroupG/H
center, socleC23×C3C22×D4
commutator, Frattini, cosocleC22×C4C24×C3
prime radicalC22×C3×D4C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23×C3 ◁ C2×C3×C4×Q8 ◁ G

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
3. C2×C3×#477C2semidirect
4. C3×#2546C2semidirect
5. C3×#2043C2semidirect
7. C3×#1985C2semidirect
9. C3×#1863C2semidirect
11. C3×#1763C2semidirect
12. #25661C3direct (N × H)
50. C2×#477C2×C3semidirect
51. #2546C2×C3semidirect
52. #2043C2×C3semidirect
54. #1985C2×C3semidirect
56. #1863C2×C3semidirect
58. #1763C2×C3semidirect
522. C3#25661direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
C2×C3×#5562
C2×C3×#4892
C2×C3×#4772
C3×#25462
C3×#20432
C3×#20422
C3×#19852
C3×#19832
C3×#18632
C3×#18622
C3×#17632
#256613